I need a sort of relay which takes only about 6V in the electromagnet, but can handle line current in the connector? It would also be preferable, not necessary, that it's inverted, so when there is power to the coil, the connector is disconnected, and when no power is in the coil, the connector is connected and 120 volts at roughly 12A flows through.

The only thing I found for this was a SPDT relay, which does not function as I would like.

Does such a thing exist? If not, what can I do to control a line power switch with a 6V circuit?

You just need a Form C relay (aka SPDT) or a Form A NC relay
A SPDT relay will do exactly what you want. Simply use the C and NC contacts instead of the C and NO contacts
Of course the "contact" side must be rated for your 120V @12A